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Properties

Label 20.0.15550983149...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{24}\cdot 5^{24}\cdot 41^{15}$
Root discriminant $256.80$
Ramified primes $2, 5, 41$
Class number $21336160$ (GRH)
Class group $[2, 2, 2, 2, 1333510]$ (GRH)
Galois group $C_5:D_5.Q_8$ (as 20T105)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![947048667401, -9288909890, 797131867410, -399721360, 304504770585, -695597980, 67166696045, -135378770, 9012577720, -12188250, 713172439, -267940, 28823490, 34040, 728915, -90, 12435, 0, 135, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 135*x^18 + 12435*x^16 - 90*x^15 + 728915*x^14 + 34040*x^13 + 28823490*x^12 - 267940*x^11 + 713172439*x^10 - 12188250*x^9 + 9012577720*x^8 - 135378770*x^7 + 67166696045*x^6 - 695597980*x^5 + 304504770585*x^4 - 399721360*x^3 + 797131867410*x^2 - 9288909890*x + 947048667401)
 
gp: K = bnfinit(x^20 + 135*x^18 + 12435*x^16 - 90*x^15 + 728915*x^14 + 34040*x^13 + 28823490*x^12 - 267940*x^11 + 713172439*x^10 - 12188250*x^9 + 9012577720*x^8 - 135378770*x^7 + 67166696045*x^6 - 695597980*x^5 + 304504770585*x^4 - 399721360*x^3 + 797131867410*x^2 - 9288909890*x + 947048667401, 1)
 

Normalized defining polynomial

\( x^{20} + 135 x^{18} + 12435 x^{16} - 90 x^{15} + 728915 x^{14} + 34040 x^{13} + 28823490 x^{12} - 267940 x^{11} + 713172439 x^{10} - 12188250 x^{9} + 9012577720 x^{8} - 135378770 x^{7} + 67166696045 x^{6} - 695597980 x^{5} + 304504770585 x^{4} - 399721360 x^{3} + 797131867410 x^{2} - 9288909890 x + 947048667401 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1555098314991537910888601000000000000000000000000=2^{24}\cdot 5^{24}\cdot 41^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $256.80$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{14} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{3}{8} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{15} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{16} a^{18} - \frac{1}{16} a^{14} - \frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} + \frac{1}{8} a^{9} - \frac{1}{16} a^{8} - \frac{1}{16} a^{6} - \frac{1}{8} a^{5} - \frac{1}{2} a^{4} + \frac{1}{8} a^{3} + \frac{1}{16} a^{2} + \frac{1}{8} a - \frac{5}{16}$, $\frac{1}{303846159705364807439374342544569864606866946828327118373811388985436034646683165305164847415114631334288288} a^{19} + \frac{6205707710026710774055556889315550320324160587731568210128228337029437644649920906011061935886116528130815}{303846159705364807439374342544569864606866946828327118373811388985436034646683165305164847415114631334288288} a^{18} - \frac{294440529162899586186187700075528468417119692140547281525927328253916995230260117152425412397086141603861}{9495192490792650232480448204517808268964592088385222449181605905794876082708848915786401481722332229196509} a^{17} + \frac{1627123314068145620090183210933141528078562915308813516150034940505649742342626447927470282278811237105113}{37980769963170600929921792818071233075858368353540889796726423623179504330835395663145605926889328916786036} a^{16} + \frac{40566418810086076164870468674974580834902619941372944505872412524281140746503717373668326697205801132717}{704979488875556397771170168316867435282753936956675448663135473284074326326411056392493845510706801239648} a^{15} + \frac{22391597790552362679404221663706790258211890524543958926373281554583708940810265105522937530016779422124059}{303846159705364807439374342544569864606866946828327118373811388985436034646683165305164847415114631334288288} a^{14} + \frac{16784635439168929642121363385461589449064277712755491009292469504287641709183034272449278305413575237435}{155658893291682790696400790237996856868272001448938072937403375504834034142768015012891827569218561134369} a^{13} + \frac{2844577068690751447114270553779282491195434474747009037952332835578258409977403344451230066039589976137063}{37980769963170600929921792818071233075858368353540889796726423623179504330835395663145605926889328916786036} a^{12} + \frac{4337810084157110390904137233796510449141065650398396176390228195809183258238879962808890834154225118772925}{151923079852682403719687171272284932303433473414163559186905694492718017323341582652582423707557315667144144} a^{11} - \frac{16370035377790865499007127834114866284628466992288757414942287438491842857031634370642149150578559990306743}{151923079852682403719687171272284932303433473414163559186905694492718017323341582652582423707557315667144144} a^{10} + \frac{16954810646851492723143713109084470882889150521824580741710784094658102862541438083239802741129229205279429}{303846159705364807439374342544569864606866946828327118373811388985436034646683165305164847415114631334288288} a^{9} + \frac{22547697783549932776867932642879481292478180896976534811874955461017220110957368156502621297290003671857889}{303846159705364807439374342544569864606866946828327118373811388985436034646683165305164847415114631334288288} a^{8} + \frac{11035443011872321423900097198863718046936845141135774660073888717508745649526558847925312614575323788911663}{303846159705364807439374342544569864606866946828327118373811388985436034646683165305164847415114631334288288} a^{7} + \frac{11353029604163053822799253947712548546519136380058029899091483271065149653510231212789034170975477366844951}{303846159705364807439374342544569864606866946828327118373811388985436034646683165305164847415114631334288288} a^{6} + \frac{68311007412977716719685992111682232510601530012687314746482637978396179748891590519731607890814670445002259}{151923079852682403719687171272284932303433473414163559186905694492718017323341582652582423707557315667144144} a^{5} - \frac{41550698436522154587977158650274576846868478575909395679169011037538256462262086528071510331884398530247337}{151923079852682403719687171272284932303433473414163559186905694492718017323341582652582423707557315667144144} a^{4} + \frac{87433211671190013642056801403906560024984923492208408961681375271881705143579279797731270949137177903207219}{303846159705364807439374342544569864606866946828327118373811388985436034646683165305164847415114631334288288} a^{3} - \frac{15186380789261770500217297112691201434652903613460416303153386425652973579768307041421983620859065570438979}{303846159705364807439374342544569864606866946828327118373811388985436034646683165305164847415114631334288288} a^{2} + \frac{16613861125878259017197208156006416606971996736569633702748259164362292916152763632284526305850910519608485}{303846159705364807439374342544569864606866946828327118373811388985436034646683165305164847415114631334288288} a - \frac{30780059364298765060442595615026995257989016631973046745737113741645281312684060630438469373462046392542975}{303846159705364807439374342544569864606866946828327118373811388985436034646683165305164847415114631334288288}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{1333510}$, which has order $21336160$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 824768892.422 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:D_5.Q_8$ (as 20T105):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $C_5:D_5.Q_8$
Character table for $C_5:D_5.Q_8$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.0.1102736.1, 10.10.452563285156250000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $20$ R $20$ $20$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.10.14.5$x^{10} - 2 x^{5} - 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
$5$5.10.14.8$x^{10} + 5 x^{8} + 20 x^{6} + 10 x^{5} + 5 x^{4} + 10 x^{2} + 12$$5$$2$$14$$F_{5}\times C_2$$[7/4]_{4}^{2}$
5.10.10.7$x^{10} + 10 x^{8} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 12$$5$$2$$10$$F_{5}\times C_2$$[5/4]_{4}^{2}$
41Data not computed